Cracking tales of historical mathematics and its interplay with science, philosophy, and culture. Revisionist history is galore. Contrarian takes on received wisdom. Implications for teaching. Informed by current scholarship. By Dr. Viktor Blåsjö.
History of mathematics research with iconoclastic madcap twists
Einstein’s theory of special relativity defines time and space operationally, that is to say, in terms of the actions performed to measure them. This is analogous to the constructivist spirit of classical geometry.
Oh no, we are chained to a wall! Aaah! This is going to mess up our geometry big time. Remember what Poincaré said: self-motion is the essence of geometry. We understand that part of the environment to be geometrical that we can cancel through self-motion, through a change of perspective.
Suppose you are looking at a chair, let’s say, and somebody tips it over so that it’s laying on its side, or somebody moves it to the other end of the room. Those are geometrical transformations: rotations and displacements in space. They are the equivalence relations of space; the isometries: things you can do without changing metric relationships.
You know that these are geometrical equivalence transformations because you can cancel them through self-motion. When the guy knocks the chair over, you can tilt your head 90 degrees, and you have restored the original visual impression of the chair. And if the guy moves the chair five meters that way, then you yourself can move five meters in the same direction and once again the chair makes precisely the same visual impression on your retina as it did before.
This is how you know that rotations and displacements are geometrical equivalence transformations. The more you accumulate experience with these kinds of scenarios, the more you begin to grasp the group of geometrical transformations as a whole. You get a global sense of what kinds of transformations are possible, how they combine and interact, and so on. This process might lead you to Euclidean or non-Euclidean conceptions of space depending on your experiences. You get to know space and what kind of geometry it has by getting to know its transformation group: that is to say, what kinds of rotations and displacements exist, what happens if you do one after the other, and so on.
Now, what about the scenario when we are chained up? We must imagine that we have been chained to this wall for life. We don’t know any other reality than this.
Our sense of what geometrical transformations are possible will be very different. There is still geometry because there are still visual impressions that we can cancel through self-motion. If an object is moving across our field of view, we can keep the retinal impressions the same by tracking it with a motion of our eyes. So we understand the geometry of sideways motion well since we can move our eyes from left to right, or point our gaze in different directions.
We also understand the geometry of depth to some extent. If an object is moving away from us, we can keep track of that through self-motion also, but of a very different kind. They eye has a lens in it. The curvature of the lens is variable and is controlled by a muscle. Depending on whether you need to focus on objects that are near or far, the muscle will pinch or pull the lens so that it is more round or more flat in order to have the right focal distance for the object you are looking at. In this way you can keep track of how much an object has moved in depth by recording how much the lens needs to be adjusted to restore focus. So this gives you the data to develop a geometry of depth.
So our chains do not deprive us of geometry altogether. We can still develop the geometry of width and the geometry of depth. But these are separate geometries to us. A free person will know that width and depth are merely two dimensions of the same kind of thing. They are both spatial dimensions. They are interchangeable and homogenous. The free person will know that since they can turn width into depth by self-motion. They just need to go stand over there and the old width is the new depth and vice versa.
But we who are chained are deprived of this experience. So to us width and depth remain qualitatively different kind of things altogether. Indeed, we measure distance in width and distance in depth completely different units. We count distance in width by the direction in which our eyes are pointing, so the unit is degrees for example. An object is 30 degrees to the left of another, for example, we might say. But we count depth by how much the lens needs to be bent to achieve focus. So the unit is something like a unit of force corresponding to the muscular effort involved. That’s a completely different kind of thing altogether, and cannot be compared with our degree measures that we used to quantify position in the width direction.
It’s not so strange that width and depth would be qualitatively different things. You already treat various measurements of the same object as qualitatively different in your everyday life. For example, suppose somebody asked you: Is this building wider than it is old? Of course that doesn’t make any sense. You cannot compare a distance in space with a duration in time. Because those quantities are determined in fundamentally different kind of ways, they are measured in completely different kinds of units, and so on. Well, just as you think time and space are not comparable, so the chained person thinks depth and width are not comparable. Samesies.
In fact, maybe you are are just as delusional as the chained guy, and for much the same reason. Actually time and space are a lot more comparable and interchangeable than you think, as Einstein’s theory of relativity says. We don’t realise this in our everyday experience, because relativistic effects become significant only at high speeds, somewhat close to the speed of light. Compared to the speed of light you have practically been standing still your whole life, even when flooring it on the highway. So you might as well have been chained to a wall. The sum total of all your visual and sensory impressions are severely and systematically impoverished just like the guy chained to a wall. Just as he doesn’t realise the fundamental unity of width and depth, so you don’t realise the fundamental unity of time and space. And for the same reason: you are both essentially standing still.
I took this example from Feynman’s famous lectures on physics. Why don’t we listen to his version as well? The classic Feynman lectures on physics are nowadays available for free at a Caltech website, audio recordings and all.
“When we look at an object, there is an obvious thing we might call the ‘apparent width’, and another we might call the ‘depth’. But the two ideas, width and depth, are not fundamental properties of the object, because if we step aside and look at the same thing from a different angle, we get a different width and a different depth, and we may develop some formulas for computing the new ones from the old ones and the angles involved. … If it were impossible ever to move, and we always saw a given object from the same position, then this whole business would be irrelevant—[width and depth] would appear to have quite different qualities, because one appears as a subtended optical angle and the other involves some focusing of the eyes …; they would seem to be very different things and would never get mixed up. It is because we can walk around that we realize that depth and width are, somehow or other, just two different aspects of the same thing.
[In Einstein’s theory of special relativity] also we have a mixture—of positions and the time. … In the space measurements of one man there is mixed in a little bit of the time, as seen by the other. Our analogy permits us to generate this idea: The ‘reality’ of an object that we are looking at is somehow greater (speaking crudely and intuitively) than its ‘width’ and its ‘depth’ because they depend upon how we look at it; when we move to a new position, our brain immediately recalculates the width and the depth. But our brain does not immediately recalculate coordinates and time when we move at high speed, because we have had no effective experience of going nearly as fast as light to appreciate the fact that time and space are also of the same nature. It is as though we were always stuck in the position of having to look at just the width of something, not being able to move our heads appreciably one way or the other.” (I.17-1)
I love this thought experiment with the chained guy. Plato’s cave 2.0. And it is perfect for our purposes today. This is going to be the concluding episode of my history and philosophy of geometry story arc, and the theme will be how everything goes full circle and the beautiful ideas from days of old are as relevant as ever to us self-absorbed moderns as well. The guy chained to a wall is a perfect backward-looking example, and a perfect forward-looking example. Back to the operationalism of Greek geometry, and forward to Einsteinian modernity.
We started, way back when, with the Greeks and their ubiquitous ruler and compass. Always with the making, those guys. Lines and circles are nothing but the things you get when you draw with these tools. Not abstract things, not axiomatically defined things. Lines and circles are operations. They are things you do.
The Greeks realised that this was the rigorous way to do mathematics. The epistemological humility of the maker is far superior to hubris of the philosopher who think they can concoct a perfect theoretical system in the abstract using the power of their mind alone. People are not as good at that as they think. Time and time again, somebody’s pretentious abstract theory has proved to contain various unintended contradictions and unnoticed assumptions. As the Greeks knew all too well: the works of Plato and Aristotle do little else than poke holes in other people’s bad theories. So we should stop trying to philosophise about essences, which we are so bad at, and instead roll up our sleeves and build stuff. Clear the junk off the table in your garage, put your tool belt on, and let’s double some cubes.
To know is to do. And this runs all the way through history. In physics, we can only know relative space, not absolute space, said Descartes and Leibniz, because we can measure the distances between things, but we cannot measure any such thing as the absolute “coordinates” of any one thing in itself without reference to anything else. So, if we believe in the virtue of the humble maker and the hubris of the speculative philosopher, then it follows that we must base our physics on relative space, not absolute space. A very reasonable conclusion, which Newton abandoned to the dismay of many at the time.
So if we stick to the classical point of view then space is what you can make, and what you can measure. What you can experience, in other words. This is a good philosophy of space. And no wonder. The Greeks, Descartes, Leibniz—back then mathematical and philosophical sophistication went hand in hand to a rare degree. So it’s no wonder they had some good ideas.
But don’t take my word for it. What makes a philosophy good? Not the say-so of some podcaster, that’s for sure. But we can prove that the classical operationalist perspective was good philosophy, by considering how it fared in the face of entirely new developments. Bad philosophy is always back-pedalling. As soon as new facts come in you have to go: uh, well, actually what I mean was… Or just descriptive: some people think they have a philosophy of something when they are just describing its basic features and making up a name for each part. But good philosophy is not that. Good philosophy is a perspective that makes you think in new ways. It gives you tools that you can use to try to understand conceptually challenging new problems. Philosophy is good if it is a fruitful way to think in challenging new situations.
Such as non-Euclidean geometry, for example. A rather counterintuitive new world; we could really use some philosophy to find our feet here. What philosophy is going to help us? Maybe Aristotle’s four different names for four different kinds of causes? Yeah right. But thinking of space and geometry in terms of operations: now that’s a philosophy. And it will prove its worth by the way it interacts with these new developments.
How do we know whether we live in a Euclidean space or a hyperbolic space? Not by developing these two geometries abstractly and axiomatically, and then testing them by their angle sum theorems or whatever. No thank you, that would be that hubristic assumption again, that we could develop geometry purely in the abstract, in the mind alone. Geometry should come from experience. But how? Modern mathematics has told us exactly how. A geometry is defined by its group of equivalence transformations, as Felix Klein said in his famous Erlanger Program. And a group of equivalence transformations can be defined in terms of experience. That is what Poincaré explained: equivalence transformations are the transformations you can cancel through self-motion. Perfect! In this way the difference between Euclidean and hyperbolic geometry emerges organically from experience itself. There is no need to postulate a hubristic ability of the human mind to develop axiomatic systems in the abstract.
Later we can go on to do more conventional abstract axiomatic mathematics as well, of course, but we do that by building on the concrete substrate developed first. We are not born with general-purpose abstract reasoning skills. We have domain-specific innate abilities such as that of acquiring a geometry by extracting the group of equivalence transformations of the space we live in from our sensory experience. And, insofar as we eventually succeed at general abstract reasoning, that is because we have mobilised our domain-specific skills and modes of thought to simulate abstract general-purpose thought. This is the point of view that I associated with Poincaré and Chomsky if you recall.
The chained guy is a perfect example to illustrate this entire tradition on geometry going all the way back to the Greeks. Restrict the operations a guy can perform, and you restrict his geometry.
I think maybe Feynman didn’t realise that his thought experiment perfectly illustrates this historically rich point of view. If we assume that Feynman came up with this through experiment himself, it seems that he started with Einstein’s relativity theory and asked himself how he could illustrate it using an analogy. Then the idea that the concepts of a physical theory depend on the kinds of experience one has, or the kinds of measurements one can make, comes off looking a bit like a kind of quirky side-effect of relativity theory. Rather than a methodological axiom built in to it from the very beginning, and indeed an axiom already strongly established long before relativity theory was even conceived.
We can see this in another one of Feynman’s remarks, in another lecture. Let’s listen to this, and pay attention to what causes what. What comes first: the physics or the philosophy?
“One of the consequences of relativity was the development of a philosophy which said, ‘You can only define what you can measure! Since it is self-evident that one cannot measure a velocity without seeing what he is measuring it relative to, therefore it is clear that there is no meaning to absolute velocity.’” (I.16-1)
One could argue that it was the other way around. This way of thinking was not a consequence of relativity theory, as Feynman says.
“One of the consequences of relativity…”
If anything, relativity theory was a consequence of this way of thinking.
“The physicists should have realized that they can talk only about what they can measure.”
Yes, they should have realized that, and they did! Not from Einstein but thousands of years before.
Indeed, Einstein read a lot of that stuff in his youth, including Ernst Mach and Poincaré. And he made no secret of how much those things influenced him. Relativity theory was a philosophy-driven scientific development to an unusual degree.
Without Poincaré’s beautiful philosophy of space, no Einstein maybe. Feynman thought the guy chained to a wall was a perfect thought experiment to describe Einstein’s theory. Yes, but it is equally perfect to describe Poincaré’s philosophy of space, which came before Einstein’s theory.
Actually Feynman was right too, when he said that the success of Einstein’s theory led to these operationalist philosophical conclusions. It did indeed. But not because science developed in its own autonomous, technical way, and then after the fact people went: huh, I wonder if we can draw some philosophical conclusion from this new science? It wasn’t like that. It was a revival of old philosophy rather than a new start stimulated by new science. Einstein’s theory didn’t so much rectify the course of philosophy, as much as it showed that the philosophers had been right all along, somewhat embarrassingly.
Remember how Newton’s absolute space was criticised. By Leibniz for example, but also others at the time. Only relative space makes epistemological sense. Only relative space is knowable. Because only relative space is measurable. Or in other words, only relative space can be operationalised.
Operationalisation is a way to ensure consistency, as the classical constructivist tradition in geometry knew. There are two ways to introduce objects in mathematics: construction, or wishlist to Santa Claus.
“Let ABC be the figure you get when you …” This is how to introduce objects by construction. “First I raise this perpendicular to that line, then I cut off a length here equal to that length over then, then I connect these two points” etc. That is the honest way to do things. The object is defined by the recipe for making it. An object is nothing but the outcome of certain operations that you perform yourself.
The other way is lazier and easier. “Let ABC be a figure such that…” This is a wishlist to Santa Claus. You state what it is that you want: “Let ABC be an equilateral triangle.” “Let ABC be a triangle with three right angles.” “Let me have a flying car and unicorn pony.” You state the properties that you want an object to have, and like a spoiled child you assume that you are thereby entitled to the object in question.
Newton was like the spoiled child asking for a unicorn. His new physics demanded absolute space and time, which were merely postulated, or wishlisted really, and cannot be operationalised.
So people like Leibniz objected, very reasonably. Newton’s physics is built on concepts that are unknowable. And it is exposed to the risk of containing inconsistencies and contradictions, since it is not susceptible to operationalisation, which has been the best way to ensure consistency since the days of Euclid. There are no unicorn ponies or triangles with three right angles, but a spoiled child wouldn’t know that, would he? Because he is not constrained by what is actually doable. So maybe Newton’s physics is ultimately incoherent since it has not taken steps to ensure otherwise.
This critique of Newton was philosophically sound, but it soon looked absolutely ridiculous. Newton’s physics was a runaway success like the world had never seen. And then you had these ridiculous little philosophers going: “well, actually, that’s actually bad science because blah blah blah.” Who would listen to such clueless nitpicks? Read the room, nerds. Newton has already won. Nobody cares about your stupid “well, actually.”
Let’s quote David Hume, for example, who was one of those philosophy losers in the 18th century. “[A] notion … beyond what we have instruments and art to make, is a mere fiction of the mind, and useless as well as incomprehensible.” Such as absolute space, for example. We have no “instruments and art”—that is to say, no physical experiments or observations—that can detect absolute space. So it must be a “useless fiction of the mind.”
Here’s another passage where Hume says the same thing: “When we entertain, therefore, any suspicion, that a philosophical term is employed without any meaning or idea (as is but too frequent), we need but enquire, from what impression is that supposed idea derived? And if it be impossible to assign any, this will serve to confirm our suspicion.” Indeed, we cannot assign any sensory impressions to the notion of absolute space, so therefore the “term is employed without any meaning.”
Hume also explains why we should insist on this criterion of meaning. “If we carry our enquiry beyond the appearances of objects to the senses, I am afraid, that most of our conclusions will be full of scepticism and uncertainty.” Meanwhile, “As long as we confine our speculations to the appearances of objects to our senses, … we are safe from all difficulties, and can never be embarrass’d by any question.” In particular, we can never run into any self-contradictions stemming from “fictions of the mind.” To be “embarrass’d” is to have contradictions in your thinking exposed to you. But just stick to the senses and “what we have instruments and art to make” and you will be fine.
Actually I kind of hate David Hume. Hume is the Galileo of philosophy: an overrated false idol who erroneously gets credit for trivial ideas that had long been obvious to mathematically and scientifically competent people.
These quotes that I just read from Hume, they are fine philosophy, or rather, they were fine philosophy a hundred years before Hume, when the same ideas were advocated by better philosophers. I have argued that those notions were well known already to Greek mathematicians. Well, we can’t prove that, because we don’t have the source evidence to know for sure one way or the other what the Greek mathematicians thought about such things. But in any case, those ideas were obviously well understood by Descartes and Leibniz for example, which is why they insisted that all geometry must be constructive, and also why they insisted that only relative space makes any philosophical sense and absolute space is a cardinal sin that must be banished from the face of the earth.
When Descartes and Leibniz said these things, they were scientifically viable ideas. Descartes and Leibniz put their money where their mouth was. They backed up their philosophy with detailed, technical scientific works, that contained both technical progress on advanced mathematical problems as well as a programmatic vision of how scientific and mathematical practice can move forward in harmony with philosophical and epistemological principles.
After Newton, that dream is dead. Philosophy lost. And all scientifically competent people knew as much. So only the scientifically ignorant, such as David Hume, kept beating this dead horse. As one historian has put it, Hume was a “dour scientific dilettante” with “almost unparalleled ignorance of the science of his day.”
Indeed, in the 18th century, only people like that could still defend this old philosophy. People with scientific integrity knew that could not in good conscience advocate for such a philosophy anymore, because that would mean that they would have to give a philosophically coherent physics that worked as well as Newton’s absolute-space-based physics, which no one could do.
So the only people who could still repeat those old philosophies that were no longer scientifically credible were now the scientifically airheaded like Hume who wouldn’t know good mathematics if it hit them in the face. That was the sad state of this once proud philosophy in the 18th century.
No wonder this anti-Newtonian philosophy became a laughing stock for centuries. And then, plot twist. They were right! Einstein’s theory exactly vindicates what these people had been saying for more than two hundred years. If you try to do science with Newton’s Santa Claus concepts of space and time, then you are doomed to run into inconsistencies. Exactly as these guys had been warning. And the way out of these hopeless inconsistencies is: operationalise everything! Exactly what these philosophy nerds had been saying all along.
Unbelievable. Imagine insisting that the most successful scientific theory of all time, that has proved itself again and again for centuries, is bound to lead to inconsistencies and self-contradictions any day now. That must be one of dumbest predictions of all time, you would think. What stubborn and oblivious people would keep embarrassing themselves by saying such silly things? And then, those guys, those very archetypes of the utter irrelevance and pointlessness of philosophy—those guys of all people—hit the cleanest home run you will ever see, with Einstein’s relativity theory. Insane.
Let’s do a bit of relativity theory here to show this. We’re on a cruise ship now. Let’s go below deck. Here there is a tennis court. Oh boy, tennis is fun! Time flies though. While we’ve been playing, has the ship reached its destination and anchored in port? Or are we still moving?
You can’t tell. That is the principle of relativity. Everything in the tennis room will the same whether the ship is moving at a constant speed or standing still. Even though one of us may be playing against the direction of travel and the other with it, that doesn’t mean that our serves and smashes will be boosted one way and slowed down the other. If we hit equal serves at the same time, they will reach the net in the center of the court at exactly the same time.
That feels natural to us in the room because we are so absorbed in the game and we are not paying any attention to whether the ship is moving or not. We are using the room as our frame of reference, or coordinate system, so to speak. That is the center of our universe at the moment.
For example, let’s say we can hit tennis serves of 50 meters per second, and it’s about 12.5 meters to the net, so it will take a quarter second for the serve to get there. That’s the science of tennis that is relevant to us when we are absorbed in the game, not whatever the ship is doing.
But the same thing works also if seen from the outside. Some guy is standing on the shore, watching us go by. And he’s a science nerd, it turns out. He happens to have one of those speed cameras that the police use to catch cars going above the speed limit.
So, according to his measurements, the ship is going 10 meters per second, and he also measures the speed of our serves somehow. Those were all 50 meters per second according to ourselves, but that’s not what the readings will say on the speed camera of the guy on the shore.
The velocities will behave additively: the speed of a projectile = the speed at which the projecter is moving + the firing speed. So when I’m serving with the ship, in the direction of travel, that’s 10+50 meters per second. So 60 is the speed measured by the observer on the shore with his speed camera. And the serve going the other way will have velocity 10-50, so 40 meters per second in the opposite direction.
So that guy disagrees with us about the speeds, but he still agrees with us that the two tennis serves will reach the net at the same time. Because the one that’s faster has further to go, since the ship is moving while the serve is in the air. Remember, it took a quarter second for the serves to reach the net. So the ship will have moved 2.5 meters in that time. So the 60 meters per second serve will have 12.5+2.5 meters to go, that’s 15 meters. And the slower serve of 40 meters per second, it has the net coming toward it so it only has to go 12.5-2.5 meters, so 10 meters before reaching the net. Indeed, going 15 meters at a speed of 60 takes the same time as going 10 meters at a speed of 40.
So the fact that the motion of the ship is undetectable to us inside the room is basically equivalent to the principle of additivity of velocities as seen from some other fixed vantage point.
Ok, yeah yeah, boring old high school physics. Who cares? That’s all trivial, right? Not really. It’s not so trivial. In fact, it’s wrong and inconsistent with other parts of physics, as we shall see.
So-called trivial things can be quite profound. We remember this from Euclid, for example. Recall for instance the construction of a square in Proposition 46. At first sight you might go: “What a big commotion about nothing. I guess Euclid wrote for kids in middle school or something. Obviously a research mathematician does not need to have squares explained to them step by step. That’s silly and trivial.”
You might have thought that, but you would have been wrong. In fact, there are no squares in spherical or hyperbolic geometry, so carefully tracking the fundamental assumptions on which the existence of squares is based is deep theoretical question. Euclid knew that. He didn’t write for middle-schoolers. He wrote for highly sophisticated mathematicians who had thought a lot about the foundations of geometry.
It’s the same with the so-called trivial physics in our tennis room. Let’s see how we can re-analyse this “trivial” situation in operationalist terms. “Let two tennis balls be fired at the same time…” Oh no, you don’t. That’s like saying “let ABCD be a square.” We can’t have that. We have to operationalise it.
It was all very easy with Newton’s absolute time, or unicorn time if you like. If this one universal absolute time is given to you by Santa Claus, then the tennis players can just use that to coordinate their serves, no problem.
But if we don’t want to rely on Santa we have to coordinate the tennis serves ourselves. Ok, so I’ll just count it off, right? “1… 2… 3… go!” Then we both serve at the same time. No, not really, because I hear “go” the instant I say it but you have to wait for the sound to travel across the room before you hear it. So we’re not really synchronised that way after all.
And it’s the same with any light-based signal of coordination, like a green light switching on, or both of us looking at the clock on the scoreboard. The speed of light is not instantaneous either, so it matters whether the sources of the signal is closer to one of us than the other.
So it looks like we are going to have to make our definition of same time depend on distance. Indeed, Einstein does precisely this. I will quote to you from the book Relativity: The Special and General Theory, of 1916. This is Einstein’s popularised presentation of his theory. Einstein says:
“We … require a definition of simultaneity such that this definition supplies us with the method by means of which, … [we] can decide by experiment whether or not [two events] occurred simultaneously. As long as this requirement is not satisfied, I allow myself to be deceived …, when I imagine that I am able to attach a meaning to the statement of simultaneity.”
Right, indeed. We can’t just say: we both checked our iPhones and it said the same time, so it was simultaneous. Who gave you those iPhones anyway? Santa Claus again. We need a do-it-yourself option. And Einstein has one for you. Here is what he says:
“[We] offer the following suggestion with which to test simultaneity [of two events, one occurring at point A and one at point B]. … The connecting line AB should be measured up and an observer placed at the mid-point M of the distance AB. This observer should be supplied with an arrangement ([such as] two mirrors …) which allows him visually to observe both places A and B at the same time. If the observer perceives the two flashes of lightning at the same time, then they are simultaneous.”
So there you have an operational definition of what it means for two things to happen at the same time. It is interesting that this operationalisation of simultaneity involved finding the midpoint of a line segment. Einstein didn’t need to explain that any further since that part was itself operationalised by Euclid, if you recall. Proposition 10 of the Elements: to find the midpoint of a given line segment.
So Einstein’s definition of simultaneity, of what it means for two things to happen at the same time, is very much in the spirit of the operationalist tradition, and the kind of physics advocated by Descartes and Leibniz. Fundamental physical notions need to be formulated in terms that show how they are knowable. Such as simultaneity being knowable or accessible to observation by defining it in terms of looking at two events using mirrors and seeing if the two events coincide or not observationally.
If it’s just about tennis, then this doesn’t really matter. That’s indeed why this way of doing physics didn’t really go anywhere for two hundred years. People just used Newton’s Santa Claus time so they didn’t have to worry about any of that stuff with the mirrors and so on.
The philosophical subtleties about what simultaneous means only become relevant at speeds comparable to the speed of light. The speed of light does not behave additively. Unlike tennis balls. If I’m standing on a moving ship and fire a tennis ball, then I add speed to the ball in addition to the speed that it already had from moving along with the ship. But if I turn on a flashlight it will go at exactly the speed of light, a natural constant, regardless of however I was traveling. It’s not the speed at which it was already going plus the new speed. It’s always the same speed of light, regardless of whether it is going with or against whatever motion of some ship or whatever.
That light behaves this way seen experimentally, not long before Einstein’s theory. The constancy of the speed of light is also embedded in the theory of electromagnetism. Maxwell’s equations of electromagnetism from the mid-19th century were hugely successful and remain a cornerstone of physics today. As Maxwell said, “light is an electromagnetic disturbance, propagated in the same medium through which other electromagnetic actions are transmitted” (Treatise on Electricity and Magnetism, 786). So light is the same kind of thing as the WiFi signal for your phone and so on.
And that was a discovery and not an assumption. As Maxwell said: “I made out the equations in the country, before I had any suspicion of the nearness between the two values of the velocity of propagation of magnetic effects and that of light.” He did that “in the country”, that is to say, at the rural family estate in the Scottish countryside. So working in isolation, in other words, and only later being able to test the theory against lab data and such things. So the fact that light can be absorbed into the theory of electromagnetism was not an assumption built in to the theory but rather something that was independently confirmed later, when Maxwell returned from “the country.”
For our purposes the point is that the constancy of the speed of light was not just an isolated experimental fact: it was a result intertwined with core physical theory already before Einstein. Indeed the title of Einstein’s famous 1905 paper on special relativity is “On the Electrodynamics of Moving Bodies”, for precisely this reason.
So it’s not so strange to give the constancy of the speed of light a big role in operationalising time and space. This axiom that the speed of light is constant is indeed built into Einstein’s definition of simultaneity. It takes light equal time to travel equal distances. If the speed of a light ray depended on the speed of the object it was coming from, then Einstein’s definition of simultaneity wouldn’t make much sense. But because of the constancy of the speed of light we can count on equal distances in equal times.
Ok, but throwing this new ingredient into the mix seems to ruin everything we said before with the tennis stuff. We already saw that the principle of relativity—that we can’t tell in the tennis room whether we are moving or not—goes hand in hand with the principle of additivity of speeds, because it was the additivity principle that made the calculations come out equivalently for different observers. So, if light speed does not behave additively, that should mean that relativity will fall as well. We should be able to exploit the constancy of the speed of light to detect the motion of the ship from inside the tennis room. Precisely what was impossible using tennis balls should now be possible using light.
Namely: You and I stand at opposite ends of the tennis court and we each have a flashlight. We turn on our flashlights at the same time, and we see which light ray reaches the net in the middle of the court first. Since the speed of light is a universal constant regardless of whether it was fired with or against the ship’s motion, the two light rays will take a different amount of time to get to the net, since the net will will have travelled some millimeters along with the ship while the light rays are in the air.
So the principle of relativity is false: it is detectable by physical experiment from within a closed room whether the ship is moving or not. Right?
No! This is precisely an example of how the naive assumptions of Newtonian physics leads to errors and inconsistencies. We turn on our flashlights “at the same time,” I said. Here I was thinking like a Newtonian. The “same time,” according to the absolute time given to us by Santa Claus. But there is no such thing. You can’t just say: let these things be done at the same time.
It is precisely by trying to operationalise the concept of time that we see how naive this Newtonian Santa Claus perspective is. From the perspective of Newtonian absolute time, either the light rays were fired at the same time or not, and either they reach the net at the same time or not. Because of absolute time, those are straightforward raw facts as it were. And they are two independent facts: fired at the same time; reaches the goal at the same time. Two separately things. Hence it makes sense to test experimentally whether those two things are coordinated or not.
But when we operationalise we see how naive we have been. As we saw above, when we followed Einstein, the only way we could define the concept of two things happening “at the same time” operationally was to say: two events happened at the same time if the light signals from them coincide when they reach the midpoint between the two locations. So we cannot independently check whether two light rays fired at the same time reach the midpoint at the same time or not. If they reach reach the midpoint at the same time, then they were fired at the same time, by definition. If they don’t reach the midpoint at the same time, they were not fired at the same time, by definition. With the humble and honest operational notion of time, that’s all we can say.
So the principle of relativity remains valid after all. There is no experiment we can do in the tennis room that shows whether the ship is moving or not. As we see when we think operationally. Despite the fact that light speed does not behave additively, which we thought was equivalent to the principle of relativity before, when we were thinking classically, in terms of Newtonian absolute time.
So how does that work in terms of the outside observer then? The guy on the shore. When we compared these two classically it was the additivity of speeds that made everything work: the speed of the ship plus the speed of the tennis serve. But now we don’t have additivity anymore since we’re dealing with light. So what happens then?
From the perspective inside the room, we fired two light rays that hit the midpoint at the same time. We concluded that they were fired simultaneously. By definition: that was the only way we were able to define the concept of simultaneity operationally.
Now, the guy outside the ship, looking at the same thing, he’s going to come to a different conclusion. Because he will have a different opinion about what the midpoint is between the two firing positions. To us on the ship, the midpoint was the net in the middle of the tennis court. But the guy on the shore thinks the net is moving, so he doesn’t think it makes any sense to use that as a reference point. Instead he will put his finger at a point that is stationary with respect to the shore. This point will initially coincide with the position of the net but as the ship moves the net will move away from this fixed point.
Because of this, events that are simultaneous as seen from within the ship are not simultaneous as seen from outside the ship. With the net on the court as the reference midpoint, the light rays arrived at the same time, and hence were by definition fired simultaneously. That was the point of view of us on the ship. But if the light rays reach that point at exactly the same moment, then they cannot also reach the different reference point selected by the outside observer at the same moment also. So they must reach that guy’s reference point at different moments, and hence they must by definition have been fired at different times, not simultaneously, according to the outside observer’s point of view.
So, when time is defined operationally, time is different to different observers. There is no God-given absolute time that tells you whether two events really happened at the same time or not. One observer thinks one thing, another observer something else. And there is no way to decide who is “right.”
If you are reading along in Einstein’s book that I mentioned, what I have just described is his section “IX. The Relativity of Simultaneity.” After that comes section “X. On the Relativity of the Conception of Distance.” If two observers disagree about time they also disagree about distance, as we shall see. In order to be able to tackle such questions we first need to define distance operationally. Let’s see how Einstein does this.
In order to do this we have to switch our imagery. Instead of people playing tennis on a ship we are going to go with lightning hitting a train, which is Einstein’s example. The thing with people playing tennis inside a ship is a good picture in some respects. Better than the train. The idea that the motion of the ship is undetectable inside the room is quite intuitive in the case of a very large ship going at a perfectly steady speed. More so than the same thing for a train, I think. Also it was nice that a tennis court has its midpoint conveniently landmarked with a net. That helped us as well to make the prominent role of the midpoint in Einstein’s definition of simultaneity more vivid in our minds, I think.
But actually this image started to break down a bit when we tried to explain the simultaneity experiment from the point of view of the guy on the shore. He was supposed to select a midpoint between the two tennis players that was stationary with respect to the shore. Which is not very practical at all. How would you hold such a point fixed, and keep your mirrors there? And besides, how does the guy on the shore determine the midpoint anyway? I mentioned that you can find the midpoint in the manner of Euclid, for instance. Fine, that works for the observer on the ship. But not the guy on the shore. He has to measure two moving targets at the same time, and somehow fix their midpoint in space at some instant, and he doesn’t even know yet what the same instant for each point even means because that is what the experiment with the mirrors placed at the midpoint is supposed to reveal.
So the ship thing doesn’t really work from a practical point of view. But we can fix these problems by switching to the train scenario instead. Alright, a train is travelling at a constant speed. It is struck by lightning at two places. Did the two lightning strikes occur at the same time or not? Well, we know how to check that. They occurred at the same time if the light from both of them reaches the midpoint at the same time. Of course, in order to check this you would have had to have your double mirrors already set up at the midpoint to begin with, as the lightnings struck. So you had to know in advance where the lightning was going to strike. But maybe that’s doable. Let’s say that the train has two lighting rods, so lightning will only strike where the lighting rods are.
In fact, we don’t really need a train as such. We can take away the walls and the roof of the train. The train is just a moving floor and it has one lightning rod at either end, and it has a pair of mirrors set up exactly halfway between them where we can see both lightning rods at the same time and judge whether the light signals from both coincide in time or not.
It doesn’t have to be lightning either. It can be a man-made thing like a light bulb, or an explosion going off. But lightning is a pretty good image in some ways. Because it conveys the idea that it is the light from the event that is the important thing, and it also conveys that the event happens at one particular instance. It is not an ongoing thing, like a light bulb staying on. It’s bam, split second, done.
And here’s another great thing about lightning that will help us a lot. It leaves a mark, a kind of burn mark, an imprint, where it struck. We don’t need that on the train, because we already have the lightning rods marking those positions. But what about the positions of the lighting strikes as seen from a stationary outsider? This gave us quite a bit of trouble with our ship example. The guy on the shore was supposed to reference things to a stationary reference frame from his point of view, but it was hard to make that concrete in that case: how do you freeze certain positions on a moving tennis court in space? You can’t. So we could only imagine that theoretically.
Now with the lightning marks this is going to work better. The train is passing a stationary train platform. A train ”station” indeed: it is stationary. When lightning strikes, it leaves a burn mark with a certain impact radius, so it also burns the platform right next to the lightning rod at that instant. Perfect! Now we have the positions where the events took place burn-marked into the stationary platform itself. So now there is no problem anymore to talk about the midpoint of the two events with respect to the stationary frame of reference. It is just the midpoint between the two burn marks. We don’t have to chase any moving targets to determine that.
So, as we said, the two observers will disagree about whether the lightning strikes were simultaneous or not. We can now quote Einstein’s explanation of this, which is the same as what I already said about the ship, but expressed in terms of the train and lightning.
“When we say that the lightning strikes A and B are simultaneous with respect to the platform, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length AB of the platform. But the events A and B also correspond to positions A and B on the train. Let M’ be the mid-point of the distance AB on the travelling train. When the flashes of lightning occur, this point M’ naturally coincides with the point M but it moves … with the velocity v of the train. If an observer sitting in the position M’ in the train did not possess this velocity, then he would remain permanently at M, and the light rays emitted by the flashes of lightning A and B would reach him simultaneously, that is, they would meet just where he is situated. Now in reality (considered with reference to the railway platform) [this observer on the train] is hastening towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A. Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A. Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash B took place earlier than the lightning flash A. We thus arrive at the important result: Events which are simultaneous with reference to the platform are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference-body (co-ordinate system) has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event.”
Right, so that’s the relativity of simultaneity, or relativity of time. Relativity is a consequence of operationalisation. And the relativity of time is in turn going to imply the relativity of space, or relativity of length, as we said. Let’s now look at that.
Let’s quote Einstein’s words here about how to compare distances across the two different reference frames. First Einstein gives a very operational definition of how distance is defined within a given reference frame, for example within the train.
“Let us consider two particular points on the train travelling along the platform with the velocity v, and inquire as to their distance apart. … An observer in the train measures the interval by marking off a measuring-rod in a straight line (… along the floor …) as many times as is necessary to take him from the one marked point to the other. Then the number which tells us how often the rod has to be laid down is the required distance.”
Great to see Einstein crawling around on all fours, laying down measuring rods. Remember Euclid in Rafael’s fresco of the School of Athens? Hunched-over with his geometry tools. We have gone full circle with Einstein. Not only does he have the same hairstyle as Rafael’s Euclid, he also has the same philosophy. If you want to understand the geometry of space you have to get operationalising, tools to the ground.
So that’s how you define distance, or operationalise distance. How many sticks long is it? How many times do I have to put a stick down to cover the whole thing, to get from one end to the other? That number is the length. And half of that number is the midpoint.
Now it gets trickier if the guy on the platform wants to measure this length. How are you supposed to do this thing with the sticks if everything is moving? One stick, two sticks, … Oh crap, the train is already long gone. That didn’t work.
The solution is to transfer out an imprint of the train, a kind of freeze-frame version of it imprinted on the platform, which we can then measure in peace and quiet after the train has left.
Kind of like the lightning marks, except the marks have to be made at the same time, and that’s precisely where it gets complicated. The same time according to whom? Were the lightning strikes simultaneous or not? Different observers disagreed about that. So when we say “make two marks on the platform corresponding to the endpoints of the train” we have to be careful about how we time the two marking events so that they are simultaneous.
Here is how Einstein puts it:
“The following method suggests itself. If we call A’ and B’ the two points on the train whose distance apart is required, then both of these points are moving with the velocity v along the platform. In the first place we require to determine the points A and B of the platform which are just being passed by the two points A’ and B’ at a particular time t, judged from the platform. These points A and B of the platform can be determined by applying the definition of time given [above]. The distance between these points A and B is then measured by repeated application of the measuring-rod along the platform. It is by no means certain that this last measurement will supply us with the same result as the [measurement performed by the observer on the train]. Thus the length of the train as measured from the platform may be different from that obtained by measuring in the train itself.”
We can picture it like this. At each end of the train there’s a guy with a paint brush. As the train passes the platform, the two guys each draw a mark on the platform at the same time. Then the length between the marks is the length of the train, surely.
But now we used the idea of “at the same time” again. So if the painters on the train draw the marks “at the same time” according to themselves, then the observer standing on the platform will go: “Noo! What are you doing?! You mistimed it! The guy at the front of the train drew his mark later than the guy at the back of the train.”
This is the direction in which the two observers disagree about simultaneity, as we saw already in the lightning examples. If the light rays from the two events to reach the midpoint of the train at the same time, then, at that moment, the guy standing at the midpoint between the two marks on the platform will already have seen the signal from the event at the back of the train, but not yet the signal from the front of the train. So simultaneous according to “train time” means that the event at the front of the train was delayed, according to “platform time.”
And the faster the train is going, the greater the delay will be. So the distance between the two painted marks will be greater for a faster train. I mean the distance between the marks drawn simultaneously according to train time. So in other words, the length of the train takes up a bigger portion of the platform. If the train covered 50% the platform when it was at rest, it will cover 80% of the platform when swishing by at a large constant speed. According to the observer on the train, that is, because we made the marks according to his simultaneity.
So “the train has grown,” you might say. Or rather, the platform has shrunk, of course. That’s how the guy in the train will interpret it. In his view, the train is the thing that remains fixed. To him the train of course has the same length all the time. For example, it takes him the same amount of effort to walk from one end of the train to the other. And it is as many sticks long, the same stick he used to measure it when the train was parked.
So when he’s swishing by the platform and makes his “simultaneous” paint marks corresponding to the length of the train, and the marks take up a greater part of the platform than before, he will say: “Huh, I guess the platform has shrunk since last time I was here.”
That is the famous length contraction phenomenon. Things that move shrink. Or rather: things that move relative to an observer appear to shrink according to that observer. The train guy thinks the platform shrunk, as we saw.
And the platform guy thinks the train shrunk, which we didn’t see directly the way we described the experiment but it must be that way because the roles of the platform and the train are interchangeable. The guy on the train can say: “I’m not moving. You’re moving!” And no physical experiment can prove him wrong. That is an axiom of relativity theory. So therefore if one guy thinks the other one shrunk, then the second one must think the first one shrunk. Because there can’t be any asymmetry, as long as they are moving at constant relative speed.
But ok, enough physics. We’re not going to become a physics podcast. My point was not to do an intro lecture on special relativity. My point was to emphasise how operational that theory is.
Time is something you do. Just as a triangle is something you do. I put a ruler on a piece of paper, I draw three intersecting lines, bam bam bam, that’s a triangle. “Oh no, that can’t be a triangle because it’s not perfect, actually. A perfect triangle must be a Platonic blah blah blah.” No. You haven’t learned the lesson of relativity theory.
Einstein didn’t use operational definitions of time and distance because he wanted his theory to be applicable in practice. He defined everything in terms of doing because the more theoretical alternative was philosophically naive and untenable.
For the same reason the Greeks reasoned in terms of constructions. Not because they were practically oriented and childlike and pre-theoretical, but precisely because they were more theoretically sophisticated, not less. Just as Einstein’s mirrors and sticks are more foundationally sophisticated than the precious abstract theory of Newton.